Shape optimization for delay of laminar-turbulent transition

Optimal control theory is applied to optimize the shape of an airfoil with respect to the energy of disturbances in the boundary layer for the purpose of delaying laminar-turbulent transition. The inviscid flow is obtained by solving the Euler equations for compressible flows, and the viscous mean flow is obtained from the solution of the boundary layer equations for compressible flows on infinite swept wings. The evolution of convectively unstable disturbances is analyzed using the linear parabolized stability equations (PSE). The results show a reduction of the total amplification of a large number of disturbances, which is assumed to represent a delay of the transition in the boundary layer. As delay of the transition implies reduction of the viscous drag, the shape optimization problem formulated here is a new approach of shape optimization to perform viscous drag reduction. The method does not intend to damp effects of the turbulence but relies on a modelling of the propagation of the disturbances in the laminar part of the boundary layer. Similar results are also obtained when simultaneously reducing the pressure drag and the disturbance kinetic energy while maintaining lift and pitch-moment coefficients near their values at initial design.